Series 6, Year 27

There are two homogenous non-rotating planets the shape of perfect sphere's with outer radii $R_{Z}$. The first of which is a perfect sphere with a density of $ρ$ a on its surface the gravitational acceleration is $a_{g}$. The second is hollow to half its radius and then its full.

• If both planets would be out of the same homogenous material, on the surface of which planet shall the gravitational acceleration be greater and what shall be the ratio of the two gravitational accelerations on the two planets?
• If the gravitational acceleration on the surface of the second planet will be $a_{g}$, what does the density of the second planet have to be?

More than a hundred years ago the measurements of surveyors confirmed that when we sail west, gravimeters show higher values of gravitational acceleration than when travelling east. Determine the difference that we measure on the equator between the measurements we make when still (relative to the earth) and when we are travelling at 20 knots per hour westwards.

Consider a copper sphere and a copper hollow shell (so thin that one can neglect its thickness). Both have the same radius at room temperature. How shall the radius change if we begin warming them up? (Find the relation between the radius and the temperature and comment on it) With the copper shell think that it has small openings which ensure that the inside and outside pressure are both the same.

In a dark corner there lurks a spider that has just caught a fly and is slowly devouring it. Assume that the consumption follows such an equation:

$$\;\mathrm{A} + \mathrm B \mathop{\rightleftharpoons}_{k_{-1}}^{k_1} \mathrm{AB} \stackrel{k_2}{\longrightarrow} \mathrm C + \mathrm B\,,$$

where A is fly substrate, B are the digestive compounds (there is always enough of themu) and C is the product of digestion. AB denotes the unstable intermediate product. The reaction is of the first order, in other words the speed is directly proportional to the concentration of the said substance. Determine how long will take the spider to digest the fly and begin hunting again, if its receptors will tell it that it is hungry once the substrate reaches 10 % of the original value.

Tip Use the approximation of the stationary state of intermediate product.

We put a roll with paper into a bearing (without friction) and we let the paper unroll itself (we neglect the sticking of layers to eac other, friction in the bearing and the weight of the bearing). What is the angular velocity of the roll after all paper is removed? We know the radiusand mass of the roll, the longitudal density of paper, its overall mass and its length. Consider that the paper shall be able to unroll into an infinite pit.

Bonus: Now consider that the paper will fall to the ground before it all unrolls.

Design a placement of lights over a table so that you will fulfill the norms for lighting. You have enough compact fluorescent lamps with a luminous flux of $P=1400lm$. Norms say that for usual work the lighting of the workplace should be $E=300lx$. The lamps can be placed into any position on the ceilling at a height of $H=2\;\mathrm{m}$ over the work desk. For simplicity's sake one can consider a square work area that has a side of $a=1\;\mathrm{m}$ a consider the lamp to be an isotropic source of light. Neglect reflection and dispersion of light.

Determine the speed of light in a translucent gelatinous cake that you will make yourself. Don't forget to describe what its composed of.

Hint: Get yourself a microwave or a laser

• How will the spectrum of an open string on a mass level $M=2⁄α′?$ How many possible states of the string on this level?
• If we consider the interaction of tachyons with other strings, we would find out, že ho můžeme popsat přibližně jako částici pohybující se v nějakém potenciálu. We consider a model of a string that is fastened on a unstable D-brane. The relevant potential of the tachyon is defined by

$$V(\phi)=\frac{1}{3\alpha'}\frac{1}{2\phi _0}(\phi-\phi _0)^2\left (\phi \frac{1}{2}\phi _0\right )\,,$$

where $$\alpha'$$

• The theory of superstrings enables the description of fermions. For their description one needs anticomutating variables. For those one creates an anticomutator instead of a comuator with the relation

$$\{A,B\}=AB BA$$

Find two such $$2\times 2$$